#285512

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,11)(3,12,9,16)(4,13,14,8)(5,15,7,10)(17,18,23,26,24,21,22,32)(19,27,28,25,20,29,31,30);(1,3,6,9)(2,4,11,14)(5,13,7,8)(10,16,15,12)(17,19,23,28,24,20,22,31)(18,25,26,29,21,30,32,27);(1,4,6,14)(2,8,11,13)(3,10,9,15)(5,12,7,16)(17,20)(18,25)(19,24)(21,30)(22,31)(23,28)(26,27)(29,32);(1,4,6,14)(2,9,11,3)(5,16,7,12)(8,15,13,10)(17,21,24,18)(19,25,20,30)(22,26,23,32)(27,28,29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192708);